Stable homotopy theory

Homotopy theory

In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space , the homotopy groups stabilize for sufficiently large. In particular, the homotopy groups of spheres stabilize for . For example, In the two examples above all the maps between homotopy groups are applications of the suspension functor. The first example is a standard corollary of the Hurewicz theorem, that . In the second example the Hopf map, , is mapped to its suspension , which generates . One of the most important problems in stable homotopy theory is the computation of stable homotopy groups of spheres. According to Freudenthal's theorem, in the the homotopy groups of spheres depend not on the specific dimensions of the spheres in the domain and target, but on the difference in those dimensions. With this in mind the k-th stable stem is . This is an abelian group for all k. It is a theorem of Jean-Pierre Serre that these groups are finite for . In fact, composition makes into a graded ring. A theorem of Goro Nishida states that all elements of positive grading in this ring are nilpotent. Thus the only prime ideals are the primes in . So the structure of is quite complicated. In the modern treatment of stable homotopy theory, spaces are typically replaced by spectra. Following this line of thought, an entire stable homotopy category can be created. This category has many nice properties that are not present in the (unstable) homotopy category of spaces, following from the fact that the suspension functor becomes invertible. For example, the notion of cofibration sequence and fibration sequence are equivalent. (Wikipedia).

Stable Homotopy Seminar, 8: The Stable Model Category of Spectra

We discuss the enrichment of spectra over spaces, and the compatibility of this enrichment with the model structure. Then we define the stable model structure by adding extra cofibrations to the levelwise model category of spectra, and restricting the weak equivalences to those maps which

From playlist Stable Homotopy Seminar

Stable Homotopy Seminar, 6: Homotopy Groups of Spectra (D. Zack Garza)

In this episode, D. Zack Garza gives an overview of stable homotopy theory and the types of problems it was designed to solve. He defines the homotopy groups of a spectrum and computes them in the fundamental case of an Eilenberg-MacLane spectrum. ~~~~~~~~~~~~~~~~======================~~~

From playlist Stable Homotopy Seminar

Stable Homotopy Seminar, 17: Universal Coefficient Theorem, Moore Spectra, and Limits

We finish constructing the universal coefficient spectral sequence, and look at some classical examples involving Moore spectra. As it turns out, it's really easy in stable homotopy theory to invert or localize at a prime. In particular, *rational* stable homotopy theory is completely alge

From playlist Stable Homotopy Seminar

Stable Homotopy Seminar, 2: Fiber and Cofiber Sequences

We review some unstable homotopy theory, especially the construction of fiber and cofiber sequences of spaces, and how they induce long exact sequences on homotopy and homology/cohomology. (There's a mistake pointed out by Jeff Carlson: when I take a CW-approximation at one point, I have

From playlist Stable Homotopy Seminar

Stable Homotopy Seminar, 7: Constructing Model Categories

A stroll through the recognition theorem for cofibrantly generated model categories, using it to construct (1) the Quillen/Serre model structure on topological spaces and (2) the levelwise model structure on spectra. The latter captures the idea that spectra are sequences of spaces, but no

From playlist Stable Homotopy Seminar

Introduction to Homotopy Theory- PART 1: UNIVERSAL CONSTRUCTIONS

The goal of this series is to develop homotopy theory from a categorical perspective, alongside the theory of model categories. We do this with the hope of eventually developing stable homotopy theory, a personal goal a passion of mine. I'm going to follow nLab's notes, but I hope to add t

From playlist Introduction to Homotopy Theory

Stable Homotopy Seminar, 4: Model categories (Ivo Vekemans)

This talk by Ivo Vekemans is a thorough introduction to model categories, presenting: weak factorization systems; the definition of model category and major examples (simplicial sets, topological spaces, and chain complexes); notions of homotopy in a model category, and the homotopy catego

From playlist Stable Homotopy Seminar

Daniel Isaksen - 1/3 Motivic and Equivariant Stable Homotopy Groups

Notes: https://nextcloud.ihes.fr/index.php/s/F2BoSJ7zgfipRxP I will discuss a program for computing C2-equivariant, ℝ-motivic, ℂ-motivic, and classical stable homotopy groups, emphasizing the connections and relationships between the four homotopical contexts. The Adams spectral sequence

From playlist Summer School 2020: Motivic, Equivariant and Non-commutative Homotopy Theory

Stable Homotopy Seminar, 1: Introduction and Motivation

We describe some features that the category of spectra is expected to have, and some ideas from topology it's expected to generalize. Along the way, we review the Freudenthal suspension theorem, and the definition of a generalized cohomology theory. ~~~~~~~~~~~~~~~~======================

From playlist Stable Homotopy Seminar

Stable Homotopy Theory by Samik Basu

PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics

From playlist Dualities in Topology and Algebra (Online)

Yonatan Harpaz - New perspectives in hermitian K-theory I

For questions and discussions of the lecture please go to our discussion forum: https://www.uni-muenster.de/TopologyQA/index.php?qa=k%26l-conference This lecture is part of the event "New perspectives on K- and L-theory", 21-25 September 2020, hosted by Mathematics Münster: https://go.wwu

From playlist New perspectives on K- and L-theory

Stable Homotopy Seminar, 14: The stable infinity-category of spectra

I give a brief introduction to infinity-categories, including their models as simplicially enriched categories and as quasi-categories, and some categorical constructions that also make sense for infinity-categories. I then describe what it means for an infinity-category to be stable and h

From playlist Stable Homotopy Seminar

Duality In Higher Categories II by Pranav Pandit

From playlist Dualities in Topology and Algebra (Online)

Spanier Whitehead Duality by Samik Basu

From playlist Dualities in Topology and Algebra (Online)

Homotopy type theory: working invariantly in homotopy theory -Guillaume Brunerie

Short talks by postdoctoral members Topic: Homotopy type theory: working invariantly in homotopy theory Speaker: Guillaume Brunerie Affiliation: Member, School of Mathematics Date: September 26, 2017 For more videos, please visit http://video.ias.edu

From playlist Mathematics

Oliver Röndigs: The slices of Hermitian K-theory (Lecture 1)

The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology: Workshop "Hermitian K-theory and trace methods" Abstract: Voevodsky constructed a filtration on the motivic stable homotopy category by measuring how many (de)suspensions with respect to the Ta

From playlist HIM Lectures: Junior Trimester Program "Topology"

Stable Homotopy Seminar, 3: The homotopy category of spectra

We discuss the Brown representability theorem, and give the Boardman-Vogt definition of the homotopy category of spectra. Examples include suspension spectra, Omega-spectra arising from cohomology theories, and Thom spectra. ~~~~~~~~~~~~~~~~======================~~~~~~~~~~~~~~~ This is

From playlist Stable Homotopy Seminar

Yonatan Harpaz - New perspectives in hermitian K-theory III

From playlist New perspectives on K- and L-theory

Stable homotopy theory

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